Algebra

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Smarandache Special Definite Algebraic Structures

In this book we introduce the notion of Smarandache special
definite algebraic structures. We can also call them equivalently
as Smarandache definite special algebraic structures. These new
structures are defined as those strong algebraic structures which
have in them a proper subset which is a weak algebraic
structure. For instance, the existence of a semigroup in a group
or a semifield in a field or a semiring in a ring. It is interesting
to note that these concepts cannot be defined when the algebraic
structure has finite cardinality i.e., when the algebraic structure
has finite number of elements in it.
Category:Algebra

K-Nomial Coefficients

In this article we will widen the concepts of "binomial coefficients" and
"trinomial coefficients" to the concept of "k-nomial coefficients", and one
obtains some general properties of these. As an application, we will
generalize the" triangle of Pascal".
Category:Algebra

Applications of Bimatrices to Some Fuzzy and Neutrosophic Models

Graphs and matrices play a vital role in the analysis
and study of several of the real world problems which
are based only on unsupervised data. The fuzzy and
neutrosophic tools like fuzzy cognitive maps invented
by Kosko and neutrosophic cognitive maps introduced
by us help in the analysis of such real world problems
and they happen to be mathematical tools which can
give the hidden pattern of the problem under
investigation. This book, in order to generalize the two
models, has systematically invented mathematical
tools like bimatrices, trimatrices, n-matrices, bigraphs,
trigraphs and n-graphs and describe some of its
properties. These concepts are also extended
neutrosophically in this book.
Category:Algebra

Introduction to Bimatrices

Matrix theory has been one of the most utilised concepts in
fuzzy models and neutrosophic models. From solving
equations to characterising linear transformations or linear
operators, matrices are used. Matrices find their applications
in several real models. In fact it is not an exaggeration if
one says that matrix theory and linear algebra (i.e. vector
spaces) form an inseparable component of each other.
Category:Algebra

Introduction to Linear Bialgebra

The algebraic structure, linear algebra happens to be one of
the subjects which yields itself to applications to several
fields like coding or communication theory, Markov chains,
representation of groups and graphs, Leontief economic
models and so on. This book has for the first time,
introduced a new algebraic structure called linear bialgebra,
which is also a very powerful algebraic tool that can yield
itself to applications.
Category:Algebra

Linear Algebra and Smarandache Linear Algebra

While I began researching for this book on linear algebra, I was a little startled.
Though, it is an accepted phenomenon, that mathematicians are rarely the ones to
react surprised, this serious search left me that way for a variety of reasons. First,
several of the linear algebra books that my institute library stocked (and it is a really
good library) were old and crumbly and dated as far back as 1913 with the most 'new'
books only being the ones published in the 1960s.
Category:Algebra

Smarandache Fuzzy Algebra

In 1965, Lofti A. Zadeh introduced the notion of a fuzzy subset of a set as
a method for representing uncertainty. It provoked, at first (and as
expected), a strong negative reaction from some influential scientists and
mathematicians - many of whom turned openly hostile. However, despite
the controversy, the subject also attracted the attention of other
mathematicians and in the following years, the field grew enormously,
finding applications in areas as diverse as washing machines to
handwriting recognition. In its trajectory of stupendous growth, it has also
come to include the theory of fuzzy algebra and for the past five decades,
several researchers have been working on concepts like fuzzy semigroup,
fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings
and so on.
Category:Algebra

Bialgebraic Structures and Smarandache Bialgebraic Structures

The study of bialgebraic structures started very recently. Till date there are no books
solely dealing with bistructures. The study of bigroups was carried out in 1994-1996.
Further research on bigroups and fuzzy bigroups was published in 1998. In the year
1999, bivector spaces was introduced. In 2001, concept of free De Morgan
bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these
bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the
construction of finite machines or finite automaton and semi automaton. The notion of
non-associative bialgebraic structures was first introduced in the year 2002. The
concept of bialgebraic structures which we define and study are slightly different from
the bistructures using category theory of Girard's classical linear logic. We do not
approach the bialgebraic structures using category theory or linear logic.
Category:Algebra

Smarandache Non-Associative Rings

An associative ring is just realized or built using reals or complex; finite or infinite
by defining two binary operations on it. But on the contrary when we want to define
or study or even introduce a non-associative ring we need two separate algebraic
structures say a commutative ring with 1 (or a field) together with a loop or a
groupoid or a vector space or a linear algebra. The two non-associative well-known
algebras viz. Lie algebras and Jordan algebras are mainly built using a vector space
over a field satisfying special identities called the Jacobi identity and Jordan identity
respectively. Study of these algebras started as early as 1940s. Hence the study of
non-associative algebras or even non-associative rings boils down to the study of
properties of vector spaces or linear algebras over fields.
Category:Algebra

Smarandache Near-Rings

Near-rings are one of the generalized structures of rings. The study and research on
near-rings is very systematic and continuous. Near-ring newsletters containing
complete and updated bibliography on the subject are published periodically by a
team of mathematicians (Editors: Yuen Fong, Alan Oswald, Gunter Pilz and K. C.
Smith) with financial assistance from the National Cheng Kung University, Taiwan.
These newsletters give an overall picture of the research carried out and the recent
advancements and new concepts in the field. Conferences devoted solely to near-rings
are held once every two years. There are about half a dozen books on near-rings apart
from the conference proceedings. Above all there is a online searchable database and
bibliography on near-rings. As a result the author feels it is very essential to have a
book on Smarandache near-rings where the Smarandache analogues of the near-ring
concepts are developed. The reader is expected to have a good background both in
algebra and in near-rings; for, several results are to be proved by the reader as an
exercise.
Category:Algebra

Smarandache Rings

Over the past 25 years, I have been immersed in research in Algebra and more
particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings)
specially to motivate both ring theorists and Smarandache algebraists to
develop and study several important and innovative properties about S-rings.
Category:Algebra

Smarandache Loops

The theory of loops (groups without associativity), though researched by several
mathematicians has not found a sound expression, for books, be it research level or
otherwise, solely dealing with the properties of loops are absent. This is in marked
contrast with group theory where books are abundantly available for all levels: as
graduate texts and as advanced research books.
Category:Algebra

Smarandache Semirings, Semifields, and Semivector Spaces

Smarandache notions, which can be undoubtedly characterized as interesting
mathematics, has the capacity of being utilized to analyse, study and introduce,
naturally, the concepts of several structures by means of extension or identification as
a substructure. Several researchers around the world working on Smarandache notions
have systematically carried out this study. This is the first book on the Smarandache
algebraic structures that have two binary operations.
Category:Algebra

Groupoids and Smarandache Groupoids

The study of Smarandache Algebraic Structure was initiated in the year 1998 by Raul
Padilla following a paper written by Florentin Smarandache called "Special Algebraic
Structures". In his research, Padilla treated the Smarandache algebraic structures mainly with
associative binary operation. Since then the subject has been pursued by a growing number of
researchers and now it would be better if one gets a coherent account of the basic and main
results in these algebraic structures. This book aims to give a systematic development of the
basic non-associative algebraic structures viz. Smarandache groupoids. Smarandache
groupoids exhibits simultaneously the properties of a semigroup and a groupoid. Such a
combined study of an associative and a non associative structure has not been so far carried
out. Except for the introduction of smarandacheian notions by Prof. Florentin Smarandache
such types of studies would have been completely absent in the mathematical world.
Category:Algebra

Smarandache Semigroups

The main motivation and desire for writing this book, is the direct appreciation
and attraction towards the Smarandache notions in general and Smarandache
algebraic structures in particular. The Smarandache semigroups exhibit properties of
both a group and a semigroup simultaneously. This book is a piece of work on
Smarandache semigroups and assumes the reader to have a good background on
group theory; we give some recollection about groups and some of its properties just
for quick reference.
Category:Algebra